Definitions
Disjoint edges in complete topological graphs
Andrew Suk
June 16, 2012
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Problem: Given a complete n-vertex simple topological graph G ,
what is the size of the largest subset of pairwise disjoint edges.
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Definition
A topological graph is a graph drawn in the plane with vertices
represented by points and edges represented by curves connecting
the corresponding points. A topological graph is simple if every
pair of its edges intersect at most once.
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Andrew Suk
Disjoint edges in complete topological graphs
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Definition
A topological graph is a graph drawn in the plane with vertices
represented by points and edges represented by curves connecting
the corresponding points. A topological graph is simple if every
pair of its edges intersect at most once.
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Andrew Suk
Disjoint edges in complete topological graphs
Definitions
We will only consider simple topological graphs.
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Disjoint edges in complete topological graphs
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Definitions
Conjecture (Conway)
Every n-vertex simple topological graph with no two disjoint edges,
has at most n edges.
Theorem (Lovász, Pach, Szegedy, 1997)
Every n-vertex simple topological graph with no two disjoint edges,
has at most 2n edges.
Best known 1.43n by Fulek and Pach, 2010.
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Andrew Suk
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Disjoint edges in complete topological graphs
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Conjecture (Conway)
Every n-vertex simple topological graph with no two disjoint edges,
has at most n edges.
Theorem (Lovász, Pach, Szegedy, 1997)
Every n-vertex simple topological graph with no two disjoint edges,
has at most 2n edges.
Best known 1.43n by Fulek and Pach, 2010.
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Andrew Suk
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Disjoint edges in complete topological graphs
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Generalization.
Theorem (Pach and Tóth, 2005)
Every n-vertex simple topological graph with no k pairwise disjoint
edges, has at most Ck n log5k−10 n edges.
Conjecture to be at most O(n) (for fixed k). By solving for k in
Ck n log5k−10 n = n2 .
Corollary (Pach and Tóth, 2005)
Every complete n-vertex simple topological graph has at least
Ω(log n/ log log n) pairwise disjoint edges.
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Conjecture (Pach and Tóth)
There exists a constant δ, such that every complete n-vertex
simple topological graph has at least Ω(nδ ) pairwise disjoint edges.
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
History
Pairwise disjoint edges in complete n-vertex simple topological
graphs:
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Ω(log1/6 n), Pach, Solymosi, Tóth, 2001.
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Ω(log n/ log log n), Pach and Tóth, 2005.
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Ω(log1+ǫ n), Fox and Sudakov, 2008.
Note ǫ ≈ 1/50. All results are slightly stronger statements.
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Main result
Theorem (Suk, 2011)
Every complete n-vertex simple topological graph has at least
Ω(n1/3 ) pairwise disjoint edges.
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Clearly the simple condition is required.
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Disjoint edges in complete topological graphs
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Clearly the simple condition is required for this problem.
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Disjoint edges in complete topological graphs
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Clearly the simple condition is required for this problem.
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Disjoint edges in complete topological graphs
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Sketch of proof
Theorem (Suk, 2011)
Every complete n-vertex simple topological graph has at least
Ω(n1/3 ) pairwise disjoint edges.
Kn+1
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Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Sketch of proof
Theorem (Suk, 2011)
Every complete n-vertex simple topological graph has at least
Ω(n1/3 ) pairwise disjoint edges.
Kn+1
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Disjoint edges in complete topological graphs
Definitions
Define F1 =
S
{Si ,j }, where Si ,j is the set of vertices inside
1≤i <j≤n
triangle v0 , vi , vj .
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S3,9 = {v1 , v5 , v7 }
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Define F1 =
S
{Si ,j }, where Si ,j is the set of vertices inside
1≤i <j≤n
triangle v0 , vi , vj .
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S3,9 = {v1 , v5 , v7 }, S2,11 = {v1 , v5 , v9 }
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Define F1 =
S
{Si ,j }, where Si ,j is the set of vertices inside
1≤i <j≤n
triangle v0 , vi , vj .
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S3,9 = {v1 , v5 , v7 }, S2,11 = {v1 , v5 , v9 }, S5,11 = {v9 }.
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
F1 is not ”complicated”.
Lemma
Any m sets in F1 , S1 , ..., Sm , partitions the ground set X into
O(m2 ) equivalence classes.
Vertices x ∼ y , if both x, y belong to the exact same sets among
S1 , ..., Sm . In other words, no set among S1 , ..., Sm contains x and
not y (and vice versa).
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Proof: m triangles partitions the plane into O(m2 ) cells.
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Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Define set system F2 =
S
1≤i <j≤n
{Si′,j }, where vk ∈ Si′,j if
topological edges v0 vk and vi vj cross.
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′ =?
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Andrew Suk
Disjoint edges in complete topological graphs
Definitions
S
Define set system F2 =
1≤i <j≤n
{Si′,j }, where vk ∈ Si′,j if
topological edges v0 vk and vi vj cross.
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′ = {v , v , v }.
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Andrew Suk
Disjoint edges in complete topological graphs
Definitions
S
Define set system F2 =
1≤i <j≤n
{Si′,j }, where vk ∈ Si′,j if
topological edges v0 vk and vi vj cross.
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′ = {v , v , v }, S ′ =?.
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Andrew Suk
Disjoint edges in complete topological graphs
Definitions
S
Define set system F2 =
1≤i <j≤n
{Si′,j }, where vk ∈ Si′,j if
topological edges v0 vk and vi vj cross.
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′ = {v , v , v }, S ′ = {v , v , v , v }.
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Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Again, F2 is not ”complicated”. Set F = F1 ∪ F2 . One can show
Lemma
Any m sets in F partitions X into at most O(m3 ) equivalence
classes.
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Main tool
Theorem (Matching theorem, Chazelle and Welzl, 1989)
Let F be a set system on an n element point set X (n is even),
such that any m sets in F partitions X into at most O(m3 )
equivalence classes. Then there exists a perfect matching M on X
such that each set in F stabs at most O(n2/3 ) members in M.
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Disjoint edges in complete topological graphs
Definitions
Main tool
Theorem (Chazelle and Welzl, 1989)
Let F be a set system on an n element point set X (n is even),
such that any m sets in F partitions X into at most O(m3 )
equivalence classes. Then there exists a perfect matching M on X
such that each set in F stabs at most O(n2/3 ) members in M.
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M = {(x1 , y1 ), (x2 , y2 ), ...., (xn/2 , yn/2 )}.
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Main tool
Theorem (Chazelle and Welzl, 1989)
Let F be a set system on an n element point set X (n is even),
such that any m sets in F partitions X into at most O(m3 )
equivalence classes. Then there exists a perfect matching M on X
such that each set in F stabs at most O(n2/3 ) members in M.
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M = {(x1 , y1 ), (x2 , y2 ), ...., (xn/2 , yn/2 )}.
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Main tool
Theorem (Matching Lemma, Chazelle and Welzl 1989)
Let F be a set system on an n element point set X (n is even),
such that any m sets in F partitions X into at most O(m3 )
equivalence classes. Then there exists a perfect matching M on X
such that each set in F stabs at most O(n2/3 ) members in M.
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M = {(x1 , y1 ), (x2 , y2 ), ...., (xn/2 , yn/2 )}.
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Auxiliary graph G , where V (G ) = M and vi vj → vk vl if Si ,j or Si′,j
stabs vk vl .
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Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Auxiliary graph G , where V (G ) = M and vi vj → vk vl if Si ,j or Si′,j
stabs vk vl .
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Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Auxiliary graph G , where V (G ) = M and vi vj → vk vl if Si ,j or Si′,j
stabs vk vl .
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Disjoint edges in complete topological graphs
Definitions
Auxiliary graph G , where V (G ) = M and vi vj → vk vl if Si ,j or Si′,j
stabs vk vl .
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Si ,j and Si′,j stabs (in total) at most O(n2/3 ) members in
M = V (G ). |E (G )| ≤ O(n5/3 ).
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
|E (G )| ≤ O(n5/3 ), by Turán, G contains an independent set of
size Ω(n1/3 ).
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Andrew Suk
Disjoint edges in complete topological graphs
Definitions
|E (G )| ≤ O(n5/3 ), by Turán, G contains an independent set of
size Ω(n1/3 ).
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Claim!
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
|E (G )| ≤ O(n5/3 ), by Turán, G contains an independent set of
size Ω(n1/3 ).
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Claim!
Andrew Suk
Disjoint edges in complete topological graphs
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Since Si ,j does NOT stab vk vl
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Assume edges cross.
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Disjoint edges in complete topological graphs
Definitions
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′ stabs v v , which is a contradiction.
Sk,l
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Disjoint edges in complete topological graphs
Definitions
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Two edges must be disjoint.
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Same argument shows
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Ω(n1/3 ) pairwise disjoint edges in Kn+1 .
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Disjoint edges in complete topological graphs
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Open Problems.
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Can the Ω(n1/3 ) bound be improved? Perhaps to Ω(n1/2 )?
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∗ (m) = Θ(m3 ).
Note that Géza Tóth show that πF
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Best known upper bound construction: O(n) pairwise disjoint
edges.
Find Ω(nδ ) pairwise disjoint edges in dense simple topological
graphs.
Andrew Suk
Disjoint edges in complete topological graphs
Definitions
Thank you!
Andrew Suk
Disjoint edges in complete topological graphs